Geometry of complex Monge-Ampère equations on compact Kähler manifolds

Abstract : In the mid 70's, Aubin-Yau solved the problem of the existence of Kähler metrics with constant negative or identically zero Ricci curvature on compact Kähler manifolds. In particular, they proved the existence and regularity of the solution of the complex Monge-Ampère equation (\omega+dd^c \varphi)^n= f\omega^n where the reference form ω is Kähler and the density f is smooth. In this thesis we look at degenerate complex Monge-Ampère equations, where the word “degenerate” stands for the fact that the reference class is merely big and not K ̈ahler or that the densities have some divisorial singularities. When looking at an equation of the type (⋆) (\theta + dd^c \varphi)^n= \mu where \mu is a positive measure, it is not always possible to make sense of the left-hand side of (⋆). It was nevertheless observed by Guedj and Zeriahi that a construction going back to Bedford and Taylor enables in this global setting to define the non-pluripolar part of the would-be positive measure (\theta+dd^c \varphi)^n for an arbitrary \theta-psh function, where \theta represents a big class. The notion of big classes is invariant by bimeromorphism while this is not the case in the Kähler setting. It is therefore natural to study the invariance property of the non-pluripolar product in the wider context of big cohomology classes. We indeed show that it is a bimeromorphic invariant. Generalizing the Aubin-Mabuchi energy functional, Boucksom, Eyssidieux, Guedj and Zeriahi introduced weighted energies associated to big cohomology classes. Under some natural assumptions, we show that such energies are also bimeromorphic invariants. We also investigate probability measures with finite energy (this concept was introduced by Berman, Boucksom, Guedj and Zeriahi) and we show that this notion is a biholomorphic but not a bimeromorphic invariant. Furtheremore, we give criteria insuring that a given measure has finite energy and test these on various examples. We then study complex Monge-Ampère equations on quasi-projective varieties. In particular we consider a compact Kähler manifold X, D a divisor and we look at the equation (\omega+dd^c \varphi)^n= f\omega^n where f is smooth outside D and with a precise behavior near the divisor. We prove that the unique normalized solution \varphi is smooth outside D and we are able to describe its asymptotic behavior near D (joint work with Hoang Chinh Lu). The solution is clearly not bounded in general and thus the idea is to find a convenient “model” function (a priori singular) bounding from below the solution. To do so we introduce generalized Monge-Ampère capacities, and use them following Kolodziej’s approach who deals with globally bounded potentials. These capacities, which generalize the Bedford-Taylor Monge-Ampère capacity, turn out to be the key point when investigating the existence and the regularity of solutions of complex Monge-Ampère equations of type (\omega+dd^c \varphi)^n= f\omega^n where f has divisorial singularities. We also treat some cases when f is not integrable, an important issue for the existence of singular Kähler-Einstein metrics on general type varieties with log-canonical singularities.
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Eleonora Di Nezza. Geometry of complex Monge-Ampère equations on compact Kähler manifolds. Differential Geometry [math.DG]. Université Toulouse III Paul Sabatier; Università degli Studi di Roma Tor Vergata, 2014. English. ⟨tel-01429436⟩

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